The Fourier inversion and the Riemann functional equation

M. Aslam Chaudhry; A. K. Al-Baiyat; B. Al-Humaidi

doi:10.1016/j.na.2010.07.037

The Fourier inversion and the Riemann functional equation

M. Aslam Chaudhry^*
, A. K. Al-Baiyat
, B. Al-Humaidi

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We prove that the Riemann functional equation can be recovered by the Mellin transforms of essentially all the absolutely integrable functions. The present analysis shows also that the Riemann functional equation is equivalent to the Fourier inversion formula. We introduce the notion of a λ-pair of absolutely integrable functions and show that the components of the λ-pair satisfy an identity involving convolution type products.

Original language	English
Pages (from-to)	3737-3741
Number of pages	5
Journal	Nonlinear Analysis, Theory, Methods and Applications
Volume	73
Issue number	12
DOIs	https://doi.org/10.1016/j.na.2010.07.037
State	Published - 15 Dec 2010

Keywords

Convolution-type products
Critical line
Critical strip
Fourier inversion formula
Functional equation
Mellin transform
Riemann hypothesis
Zeta function

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1016/j.na.2010.07.037

Cite this

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title = "The Fourier inversion and the Riemann functional equation",

abstract = "We prove that the Riemann functional equation can be recovered by the Mellin transforms of essentially all the absolutely integrable functions. The present analysis shows also that the Riemann functional equation is equivalent to the Fourier inversion formula. We introduce the notion of a λ-pair of absolutely integrable functions and show that the components of the λ-pair satisfy an identity involving convolution type products.",

keywords = "Convolution-type products, Critical line, Critical strip, Fourier inversion formula, Functional equation, Mellin transform, Riemann hypothesis, Zeta function",

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T1 - The Fourier inversion and the Riemann functional equation

AU - Aslam Chaudhry, M.

AU - Al-Baiyat, A. K.

AU - Al-Humaidi, B.

PY - 2010/12/15

Y1 - 2010/12/15

N2 - We prove that the Riemann functional equation can be recovered by the Mellin transforms of essentially all the absolutely integrable functions. The present analysis shows also that the Riemann functional equation is equivalent to the Fourier inversion formula. We introduce the notion of a λ-pair of absolutely integrable functions and show that the components of the λ-pair satisfy an identity involving convolution type products.

AB - We prove that the Riemann functional equation can be recovered by the Mellin transforms of essentially all the absolutely integrable functions. The present analysis shows also that the Riemann functional equation is equivalent to the Fourier inversion formula. We introduce the notion of a λ-pair of absolutely integrable functions and show that the components of the λ-pair satisfy an identity involving convolution type products.

KW - Convolution-type products

KW - Critical line

KW - Critical strip

KW - Fourier inversion formula

KW - Functional equation

KW - Mellin transform

KW - Riemann hypothesis

KW - Zeta function

UR - https://www.scopus.com/pages/publications/77956907239

U2 - 10.1016/j.na.2010.07.037

DO - 10.1016/j.na.2010.07.037

M3 - Article

AN - SCOPUS:77956907239

SN - 0362-546X

VL - 73

SP - 3737

EP - 3741

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

IS - 12

ER -

The Fourier inversion and the Riemann functional equation

Abstract

Keywords

ASJC Scopus subject areas

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